Measures of Dispersion Dr. R Jayashree Veterinary College Shimoga
Central Tendency and Variation • Central tendency: mode, median, mean • Variation: range, average(mean) deviation, variance, and standard deviation
The idea is the figure out what is happening with your group
STAT 512 2018-19 VCS AGB RJ
Central Tendency • Statisticians need a way to indicate where, along a scale of measures, a given distribution is centered. • They used three measures of central tendency – The Mode – The Median – The Mean STAT 512 2018-19 VCS AGB RJ
Measure of Dispersion Measures of dispersion are the measures of “spread” or “scatteredness” of observations about an average.
STAT 512 2018-19 VCS AGB RJ
Measure of Dispersion • The term dispersion is used to indicate the fact that within a given group of data, the observations differ from one another in value or that there is lack of uniformity in their size.
STAT 512 2018-19 VCS AGB RJ
Measure of Dispersion The degree to which numerical; data tend to spread about an average or central value is called the variation or dispersion of data. Correlation visualized programmed by Erich Neuwirth target value
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empirical (data) value
0.052226 2
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STAT 512 2018-19 VCS AGB RJ
Measures of dispersion help to study the important characteristics of the dispersion, by measuring the extents to which there are differences between individual observations and the measure of central tendency.
STAT 512 2018-19 VCS AGB RJ
The various measures of dispersion are 1. 2. 3. 4.
Range Variance Standard deviation Coefficient of variation.
STAT 512 2018-19 VCS AGB RJ
Range It is defined as the difference between the highest and the lowest values for the variable in distribution denoted by ‘R’. Range= Highest value- Lowest value. In case of frequency distribution it is calculated as the difference between the upper limit of the highest class and the lower limit of the smallest class. STAT 512 2018-19 VCS AGB RJ
Coefficient of range • It is a relative measure of the range and estimated by using the formula: Coefficient of Range=
L_S X100 L+S
L= Highest value in the series S= Smallest value in the series. STAT 512 2018-19 VCS AGB RJ
Variance R.A.Fisher first introduced the term in 1913. • Variance is the mean of the squared deviations about the mean of a series. Variance is the square of standard deviation • It is denoted by S2 for sample variance • σ2 for population.
STAT 512 2018-19 VCS AGB RJ
Computation of variance Direct method: S2= ∑(X- X)2 n-1 Calculator of Machine formula: S2=∑X2-∑(X)2 n n-1 STAT 512 2018-19 VCS AGB RJ
Standard Deviation • Standard deviation for population is denoted by ‘σ ’ and for sample as ‘s’. • This was first suggested by Karl Pearson in 1893. • It is defined as the positive square root of the mean of the squares of deviations of the given observations from their arithmetic mean. STAT 512 2018-19 VCS AGB RJ
Coefficient of variation It is a relative measure of dispersion. Coefficient of variation is the percentage variation in mean, standard deviation being considered as the total variation in the mean and symbolically denoted as C.V. or It is a ratio between standard deviation and mean of sample expressed in percentage. STAT 512 2018-19 VCS AGB RJ
Coefficient of variation s CV= × 100 X S= Standard deviation of the sample X = mean of the sample
STAT 512 2018-19 VCS AGB RJ
The following are the haemoglobin values 9g/100ml) of piglets receiving treatment for piglet anaemia: 9.1,10,11.4,12.4,9.8,8.3,9.9,9.1,7.5,6.7 . Calculate the sample mean, variance, standard deviation and coefficient of variation. STAT 512 2018-19 VCS AGB RJ
X 9.1 10 11.4 12.4 9.8 8.3 9.9 9.1 7.5 6.9 ∑X=94.2
X2 82.81 100 129.96 153.76 96.04 68.89 98.01 82.81 56.25 44.89 ∑X2=913.89
STAT 512 2018-19 VCS AGB RJ
Mean= ∑X
=
n Variance=
S2
94.2 =
9.42
10 =
(∑X)2
∑X2
913.42- 8873.6 10
n
n-1 Standard Deviation=
= 2.90
10-1
2.90
=1.70
Coefficient of variation (C.V)= S
X 100
X =
1.70 X 100 = 18.046% 9.42
STAT 512 2018-19 VCS AGB RJ
Formulas for variability Note the similarity of these formulas
AD
Average deviation
Variance
2
Standard deviation
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X
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2
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X
STAT 512 2018-19 VCS AGB RJ
N
2